Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Efficiency and Bonus Hunger 257
likelihood maximization. Note however that Cooray & Ananda (2005) did not consider the
case where explanatory variables were available, so that their approach has to be extended
to this more realistic situation.
5.5.4 Alternative Approaches to Risk Classification
There are numerous techniques applied to the modelling of insurance losses. Early references
in actuarial science include ter Berg (1980a,b). Various mathematical and statistical models
for estimation of automobile insurance pricing are reviewed in Weisberg, Tomberlin &
Chatterjee (1984). The methods are compared on their predictive ability based on two sets
of automobile insurance data for two different states collected over two different periods.
The issue of model complexity versus data availability is resolved through a comparison of
the accuracy of prediction. The models reviewed range from the use of simple cell means
to various multiplicative-additive schemes to the empirical Bayes approach. The empirical
Bayes approach, with prediction based on both model-based and individual cell estimates,
seems to yield the best forecast. See also Jee (1989).
Williams & Huang (1996) applied KDD (for Knowledge Discovery in Databases)
techniques for insurance risk assessment. Daengdej, Lukose & Murison (1999) considered
CBR (for Case-Based Reasoning) techniques for claim predictions.
Classification techniques are also often used for risk classification. Retzlaff-Roberts &
Puelz (1996) adopted an efficiency approach to the two-group linear programming method
of discriminant analysis, using principles taken from data envelopment analysis, to predict
group membership in an insurance underwriting scheme. Yeo ET AL. (2001) applied clustering
techniques before modelling insurance losses.
5.5.5 Efficiency
The efficiency (or elasticity) of a bonus-malus system was first studied by Loimaranta
(1972) and De Pril (1978). Note however that in these papers, the unknown individual risk
factors are not viewed as random variables. These authors work in the fixed effects model
that is close in spirit to the limited fluctuations credibility theory. Their efficiency concepts
are therefore entirely different from the notion of efficiency proposed in Norberg (1976).
Other efficiency measures have been proposed in the literature. For instance, Heras ET AL.
(2002) evaluated the asymptotic fairness of bonus-malus systems (i.e., their ability to assess
the individual risk in the long run) assuming the simplest case when there is no hunger for
bonus.
5.5.6 Optimal Retention Limits and Bonus Hunger
The problem of determining the optimal claim size has been the topic of several papers.
De Leve & Weeda (1968) considered a −1/top bonus-malus scale, so that the decision to
file or not has to be made only if no claim has been made during the same period. Lemaire
(1976,1977) studied the hunger for bonus and proposed a dynamic programming algorithm
to determine the optimal claiming behaviour. De Pril (1979) considered that the claims
were generated by a Poisson process and adapted a continuous-time approach. Specifically,